Algebra

PrepBank · পাতা ৫ / ১৪ · ৪০১–৫০০ / ১,৩৮০

৪০১.

Which of the following is irrational?

ক
3/5
খ
√2
গ
0.75
ঘ
1.2

ব্যাখ্যা

Question: Which of the following is irrational?

Solution:
অমূলদ সংখ্যা (irrational number):
- যে সংখ্যাকে p/q আকারে প্রকাশ করা যায় না, যেখানে p ও q পূর্ণসংখ্যা এবং q ≠ 0, সে সংখ্যাকে অমূলদ সংখ্যা বলা হয়।
- পূর্ণবর্গ নয় এরূপ যে কোনো স্বাভাবিক সংখ্যার বর্গমূল কিংবা তার ভগ্নাংশ একটি অমূলদ সংখ্যা। যেমন, √2 = 1.414213..., √6 = 2.229489... ইত্যাদি অমূলদ সংখ্যা।
- কোনো অমূলদ সংখ্যাকে দুইটিপূর্ণ সংখ্যার অনুপাত হিসেবে প্রকাশ করা যায় না।
-অমূলদ সংখ্যাকে একটি মূলদ সংখ্যা দ্বারা গুণ করলে অমূলদ সংখ্যা পাওয়া যায়।
অর্থাৎ, non zero rational number × irrational number = irrational number.

অপশন গুলো ব্যাখ্যা করে,
ক) 3/5 ; এটি একটি ভগ্নাংশ, তাই rational
খ) √2 ; এটি কোন পূর্ণসংখ্যার বর্গমূল নয়, তাই irrational
গ) 0.75 = 3/4 ; ভগ্নাংশের সমান, তাই rational
ঘ) 1.2 = 6/5 ; ভগ্নাংশের সমান, তাই rational

সুতরাং, √2 একটি অমূলদ সংখ্যা।

৪০২.

Which number will continue the following series? 3, 8, 15, 24, 35, 48, 63, 80 ___

ক
ক) 92
খ
খ) 91
গ
গ) 95
ঘ
ঘ) 97
ঙ
ঙ) 99

ব্যাখ্যা

ধারাটিঃ
3 + 5 = 8
8 + 7 = 15
15 + 9 = 24
24 + 11 = 35
35 + 13 = 48
48 + 15 = 63
63 + 17 = 80
80 + 19 = 99

৪০৩.

ক
0
খ
11
গ
9
ঘ
5

ব্যাখ্যা

Question:

Solution:

৪০৪.

ক
8
খ
16
গ
36
ঘ
9
ঙ
None

ব্যাখ্যা

Question:

Solution:

৪০৫.

If x = 2 + √3 and y = 2 - √3, find the value of (x² + y²)²

ক
169
খ
196
গ
16
ঘ
144

ব্যাখ্যা

Question: If x = 2 + √3 and y = 2 - √3, find the value of (x² + y²)²

Solution:
We are given:
x = 2 + √3, y = 2 - √3

Now,
x + y
= (2 + √3) + (2 - √3)
= 4

And,
⇒ xy
= (2 + √3)(2 - √3)
= 2² - (√3)²
= 4 - 3 = 1

We know,
x² + y² = (x + y)² - 2xy
⇒ x² + y² = (4)² - 2(1) [Substitute the values]
⇒ x² + y² = 16 - 2
⇒ x² + y² = 14

∴ (x² + y²)² = 14² = 196

৪০৬.

If -1 ≤ 3 - 2x ≤ 3, then -

ক
ক) 0 ≤ x ≤ 2
খ
খ) -2 ≤ x ≤ 0
গ
গ) x ≤ 2
ঘ
ঘ) x ≥ 0

ব্যাখ্যা

Given, -1 ≤ 3 - 2x ≤ 3
⇒ -4 ≤ - 2x ≤ 0 [Adding - 3 in every section]
⇒ 2 ≥ x ≥ 0
⇒ 0 ≤ x ≤ 2

৪০৭.

For a geometric sequence, the first term a = 7 and the common ratio r = 2 . What is the sum of the first 5 terms?

ক
264
খ
225
গ
217
ঘ
198

ব্যাখ্যা

Question: For a geometric sequence, the first term a = 7 and the common ratio r = 2 . What is the sum of the first 5 terms?

Solution:
Given,
a = 7
r = 2 > 1
n = 5

S = {a(rⁿ - 1)}/(r - 1)
= {7(2⁵ - 1)}/(2 - 1)
= 7 × 31
= 217

৪০৮.

If 2√(2x + 1) + 5 = 8, then x =

ক
ক) 5/4
খ
খ) 2.5
গ
গ) 5/8
ঘ
ঘ) 5

ব্যাখ্যা

Question: If 2√(2x + 1) + 5 = 8, then x =

Solution:
Given that,
2√(2x + 1) + 5 = 8
⇒ 2√(2x + 1) = 3
⇒ √(2x + 1) = 3/2
⇒ 2x + 1 = 9/4
⇒ 2x = (9/4) - 1
⇒ 2x = 5/4
∴ x = 5/8

৪০৯.

If a set has 5 elements, then the power set of that set has ______ elements.

ক
5
খ
125
গ
10
ঘ
32

ব্যাখ্যা

Question: If a set has 5 elements, then the power set of that set has ______ elements.

Solution:
The number of elements in the power set of a set with n elements is 2ⁿ.
Here, n = 5
∴ Number of elements in the power set = 2⁵ = 32

So the power set has 32 elements.

৪১০.

Find the determinant of the matrix:

ক
2
খ
4
গ
- 2
ঘ
- 6

ব্যাখ্যা

Question: Find the determinant of the matrix:

Solution:
2 × 2 ম্যাট্রিক্সের জন্য:

⇒ det(A) = ad - bc

Given matrix,

∴ det(A) = 1 × 4 - 2 × 3 = 4 - 6 = - 2

৪১১.

What will come at the place of question mark?
6, 24, 60, 120, 210, ?

ক
336
খ
343
গ
350
ঘ
380

ব্যাখ্যা

Question: What will come at the place of question mark?
6, 24, 60, 120, 210, ?

Solition:
2³ - 2 = 6
3³ - 3 = 24
4³ - 4 = 60
5³ - 5 = 120
6³ - 6 = 210
7³ - 7 = 336

৪১২.

In a survey of 100 people, 60 people like coffee, 50 people like tea, and 20 people like both coffee and tea. How many people like only coffee?

ক
10
খ
20
গ
30
ঘ
40
ঙ
60

ব্যাখ্যা

Question: In a survey of 100 people, 60 people like coffee, 50 people like tea, and 20 people like both coffee and tea. How many people like only coffee?

Solution:
Total number of people surveyed = 100
People who like coffee = 60
People who like tea = 50
People who like both coffee and tea = 20
People who like only coffee can be calculated as follows:

People who like only coffee = People who like coffee - People who like both coffee and tea
= 60 - 20
= 40

Therefore, the number of people who like only coffee is 40.

৪১৩.

If x² - 6x + 8 < 0, then solve the inequality.

ক
x < 2 or x > 4
খ
2 ≤ x ≤ 4
গ
2 < x < 4
ঘ
x > 4

ব্যাখ্যা

Question: If x² - 6x + 8 < 0, then solve the inequality.

Solution:
x² - 6x + 8 < 0
⇒ x² - 2x - 4x + 8 < 0
⇒ x(x - 2) - 4(x - 2) < 0
⇒ (x - 2)(x - 4) < 0

দুটি রাশির গুণফল শূন্যের চেয়ে ছোট হওয়ার শর্ত হলো, একটি রাশি ধনাত্মক এবং অন্যটি ঋণাত্মক হতে হবে।
∴ অসমতাটি সত্য হবে যদি x - 2 > 0 এবং x - 4 < 0 হয়।

x - 2 > 0 অর্থাৎ, x > 2
x - 4 < 0 অর্থাৎ, x < 4

∴ অসমতার সমাধান হলো 2 < x < 4

৪১৪.

If m and n are in the domain of the function g and g(m) > g(n), which of the following must be true?

ক
ক) Mn ≠ 0
খ
খ) M > n
গ
গ) M < n
ঘ
ঘ) m ≠ n

ব্যাখ্যা

If g(m) > g(n)
∴ m > n
So, m ≠ n

৪১৫.

The factor of x² - 5x - 6 are:

ক
ক) (x - 6)(x + 1)
খ
খ) (x + 6)(x - 1)
গ
গ) (x - 3)(x + 2)
ঘ
ঘ) (x - 3)(x - 2)

ব্যাখ্যা

x² - 5x - 6
x² - 6x + x - 6
x(x - 6) + 1(x - 6)
(x - 6)(x + 1).

৪১৬.

If x - y = 3 then what is the value of x³ - y³ − 9xy?

ক
ক) 12
খ
খ) 18
গ
গ) 20
ঘ
ঘ) 27

ব্যাখ্যা

x³ - y³ − 9xy
= (x - y)³ + 3xy(x - y) - 9xy
= (3)³ + 9xy - 9xy
= 27

৪১৭.

If α and β are roots of the equation x² + x - 1 = 0, then the equation whose roots are α/β and β/α is:

ক
ক) x² + 3x + 1 = 0
খ
খ) x² - 3x + 1 = 0
গ
গ) x² + 3x - 1 = 0
ঘ
ঘ) 2x² - 3x + 1 = 0

ব্যাখ্যা

Question: If α and β are roots of the equation x² + x - 1 = 0, then the equation whose roots are α/β and β/α is:

Solution:
As α and β are roots of x² + x - 1 = 0,
then
⇒ α + β = - ( + 1) = - 1
⇒ αβ = - 1

Now, if (α/β) and (β/α) are roots then,
⇒ Sum of roots = (α/β) + (β/α)
= (α² + β²)/αβ
= [(α + β)² - 2αβ]/αβ
= (- 1)² - 2(-1)]/(-1)
= (1+ 2)/(- 1)
= - 3

Product of roots = (α/β) × (β/α) = 1
Now, then the equation is,
⇒ x² - (Sum of roots)x + Product of roots = 0
⇒ x² - (- 3)x + (1) = 0
⇒ x² + 3x + 1 = 0

৪১৮.

If a+b = 6 and ab = 16, what is the value of a²+b²?

ক
ক) 14
খ
খ) 9
গ
গ) 4
ঘ
ঘ) 30

ব্যাখ্যা

দেওয়া আছে, a+b = 6, ab = 16
আমরা জানি, a²+b² = (a+b)²-2ab
= 6²-2×16 = 4

৪১৯.

Quantity in A = (9/13)² and Quantity B = (9/13)^1/2

ক
ক) Quantity A equals Quantity B
খ
খ) Relationship indeterminate.
গ
গ) Quantity B is greater
ঘ
ঘ) Quantity A is greater
ঙ
ঙ) None of these

ব্যাখ্যা

Question: Quantity in A = (9/13)² and Quantity B = (9/13)^1/2

Solution:
Quantity in A = (9/13)²
A² = (9/13)²
= 81/169

B = (9/13)^1/2
B² = 9/13
= (9 × 13)/(13 × 13)
= 117/169
A² < B²

∴ Quantity B > Quantity A

৪২০.

The solution of 2x² + 3x - 2 = 0 are

ক
- 2 and 1/2
খ
2 and 1/2
গ
1 and 1/2
ঘ
- 2 and - 1/2

ব্যাখ্যা

Question: The solution of 2x² + 3x - 2 = 0 are

Solutiuon:
Given,
2x² + 3x - 2 = 0
⇒ 2x² + 4x - x - 2 = 0
⇒ 2x(x + 2) - 1(x + 2) = 0
⇒ (x + 2) (2x - 1) = 0

So, x + 2 = 0
x = - 2

Or, 2x - 1 = 0
∴ x = 1/2

৪২১.

If k, 2k - 1 and 2k + 1 are three consecutive terms of an AP, the value of k is

ক
1
খ
2
গ
3
ঘ
- 4
ঙ
None of the above

ব্যাখ্যা

Question: If k, 2k - 1 and 2k + 1 are three consecutive terms of an AP, the value of k is

Solution:
Here, k, 2k - 1 and 2k + 1 are three terms in A.P.

Then,
The common difference between the first two terms is = 2k - 1 - k = k - 1
The common difference between the next two terms is = 2k + 1 - 2k + 1 = 2

Hence,
k - 1 = 2
∴ k = 3

৪২২.

If 1/x = 7/3 then 17/(x + 2) = ?

ক
ক) 6
খ
খ) 7
গ
গ) 17
ঘ
ঘ) 1

ব্যাখ্যা

দেয়া আছে,
1/x = 7/3
x = 3/7

17/(x + 2) = 17/{(3/7) + 2}
= 17/{(3 + 14)/7}
= 17/(17/7)
= 17 × (7/17)
= 7

৪২৩.

If f(x) = x + 5 and g(x) = x - 5 then, f(g(x))=?

ক
0
খ
1
গ
x
ঘ
5

ব্যাখ্যা

প্রশ্ন: If f(x) = x + 5 and g(x) = x - 5 then, f(g(x))=?

সমাধান:
g(x) = x - 5
f(x) = x + 5
f(g(x)) = g(x) + 5
= x - 5 + 5
= x

৪২৪.

If (3x + 5y)/(3x - 5y) = 4; what is the value of x/y?

ক
ক) 9/25
খ
খ) 25/3
গ
গ) 25/9
ঘ
ঘ) 25/16

ব্যাখ্যা

Question: If (3x + 5y)/(3x - 5y) = 4; what is the value of x/y?

Solution:
given,
(3x + 5y)/(3x - 5y) = 4
3x + 5y = 4(3x - 5y)
3x + 5y = 12x - 20y
5y + 20y = 12x - 3x
25y = 9x
x/y = 25/9

৪২৫.

Which of the following is an equation which graph is a set of points equidistant from the points {0, 0} and {6, 0}?

ক
ক) x = 3
খ
খ) y = 3
গ
গ) x = 3y
ঘ
ঘ) y = 3x

ব্যাখ্যা

The points equidistant from the point (0, 0) and (6, 0) will be = {(0 + 6)/2, (0 + 0)/2}
= (3, 0)
∴ x = 3, y = 0
So, the answer will be x = 3

৪২৬.

If (x/y) + (y/x) = √7 then what is the value of [(x⁴/y⁴) + (y⁴/x⁴)]³ ?

ক
12,167
খ
12,317
গ
21,167
ঘ
12,377

ব্যাখ্যা

Question: If (x/y) + (y/x) = √7 then what is the value of [(x⁴/y⁴) + (y⁴/x⁴)]³ ?
(Janata RC 22 এর অনুরূপ)

Solution:
দেওয়া আছে,
(x/y) + (y/x) = √7

∴ x⁴/y⁴ + y⁴/x⁴
= (x/y)⁴ + (y/x)⁴
= {(x/y)²}² + {(y/x)²}²
= {(x/y)² + (y/x)²}² - 2.(x²/y²).(y²/x²)
= {(x/y)² + (y/x)²}² - 2
= [{(x/y) + (y/x)}² - 2.(x/y).(y/x)]² - 2
= {(√7)² - 2}² - 2
= (7 - 2)² - 2
= 5² - 2
= 25 - 2
= 23

[(x⁴/y⁴) + (y⁴/x⁴)]³ = (23)³ = 12,167

৪২৭.

The value of is:

ক
0
খ
1
গ
1/2
ঘ
2

ব্যাখ্যা

Question: The value ofis:
(Officer General 22 এর অনুরূপ)

Solution:
1 - cosx
≈ x²/2 for small x

Again,
1 - cosx/x²= (x²/2)/x²= 1/2

৪২৮.

If x is a whole number greater than 1, then x²(x² - 1) is always divisible by?

ক
ক) 8
খ
খ) 10
গ
গ) 12
ঘ
ঘ) 16

ব্যাখ্যা

1এর চেয়ে বড় সংখ্যা 2
ধরি
x = 2
প্রদত্ত রাশিমালা = x²(x² - 1)
= 2²(2² - 1)
= 4(4 - 1)
= 4 × 3
= 12
x²(x² - 1) রাশিটি সর্বদা 12 দ্বারা বিভাজ্য হবে।

৪২৯.

What is the 8^th term of the series: 2 + 4 + 8 + 16 +..........

ক
128
খ
256
গ
512
ঘ
1024

ব্যাখ্যা

Question: What is the 8^th term of the series: 2 + 4 + 8 + 16 +..........

Solution:
Given,
a = 2
r = 4/2
= 2 > 1
The given series is a geometric series.

We know,
the n^th term of a geometric series = ar^{(n - 1)}

8^th term of the series = 2 × 2^{(8 - 1)} = 2 × 2⁷ = 256

৪৩০.

If x + 2y = 4 and x/y = 2, then determine the value of x and y.

ক
1,2
খ
2,3
গ
2,1
ঘ
3,4

ব্যাখ্যা

দেয়া আছে
x + 2y = 4..............(1)
এবং
x/y = 2
x = 2y...............(2)

x এর মান (1) নং সমীকরণে বসিয়ে পাই
x + 2y = 4
2y + 2y = 4
4y = 4
y = 1

y এর মান (2) নং সমীকরণে বসিয়ে পাই
x = 2y
x = 2 × 1 = 2

নির্ণেয় সমধান (x, y) = (2,1)

৪৩১.

If a² + b² = 5ab then find the value of (a²/b² + b²/a²)

ক
ক) 32
খ
খ) 16
গ
গ) 23
ঘ
ঘ) -23

ব্যাখ্যা

a² + b² = 5ab
a²/ab + b²/ab = 5
a/b + b/a = 5
Squaring the both sides
(a/b)² + (b/a)² = (5)²
a²/b² + b²/a² + 2 × (a/b) × (b/a)= 25
a²/b² + b²/a² + 2 = 25
a²/b² + b²/a² = 25 -2
a²/b² + b²/a² = 23.

৪৩২.

What is the next term in the sequence - 4, 9, 6, 11, 8, 13, …… ?

ক
ক) 18
খ
খ) 10
গ
গ) 16
ঘ
ঘ) 9

ব্যাখ্যা

এখানে দুটি ধারা বিদ্যমান।
বিজোড় অবস্থানের ধারা = 4, 6, 8, 10
জোড় অবস্থানের ধারা = 9, 11, 13, 15

৪৩৩.

The average pocket money of 3 friends Sharifullah, Nawshad, Tuhin is Tk. 80 in a particular month. If Nawshad spends double and Tuhin spends triple of what Sharifullah spends during that month and if the average of their unspent pocket money is Tk. 60, then Tuhin spends -

ক
Tk. 10
খ
Tk. 12
গ
Tk. 30
ঘ
Tk. 36

ব্যাখ্যা

Question: The average pocket money of 3 friends Sharifullah, Nawshad, Tuhin is Tk. 80 in a particular month. If Nawshad spends double and Tuhin spends triple of what Sharifullah spends during that month and if the average of their unspent pocket money is Tk. 60, then Tuhin spends -

Solution:
Total pocket money = 3 × 80
= 240 tk

total unspent money = 3 × 60
= 180 tk

Total spent money = 240 - 180 tk
= 60 tk

let, sharifullah spends x tk
nawshad spends 2x tk
tuhin spends 3x tk

x + 2x + 3x = 60
⇒ 6x = 60
∴ x = 10 taka

Tuhin spends = 3 × 10 taka
= 30 taka

৪৩৪.

If the median of 21 observations is 40 and if the observations greater than the median are increased by 5 then the median of the new data will be-

ক
45 - (50/21)
খ
45
গ
40
ঘ
40 + (50/21)

ব্যাখ্যা

Question: If the median of 21 observations is 40 and if the observations greater than the median are increased by 5 then the median of the new data will be-

Solution:
• The median of a set of data is the middle value when the data is arranged in ascending order.

For 21 observations, the median is the 11th value.
According to the question, the original median is 40. Now, observations greater than the median are increased by 5. Since only the values greater than the median are increased, the position of the median (the 11th observation) remains unchanged, and the value of the median (40) itself does not change.

⇒ The median of the new data will still be 40.
∴ The median of the new data remains 40.

৪৩৫.

If x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0, then, find the value of x³⁵ + x⁶³.

ক
ক) 0
খ
খ) 1
গ
গ) 2
ঘ
ঘ) 4

ব্যাখ্যা

Question: If x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0, then, find the value of x³⁵+ x⁶³.

Solution:
Given that
x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0
x⁶ + x⁵ + x⁴ + x³ + x² + x = - 1..........(1)

Multiplying by x,
we have
x⁷+ x⁶ + x⁵ + x⁴ + x³ + x² + x = 0 ........(2)
⇒ x⁷ - 1 = 0
⇒ x⁷ = 1

According to question
x³⁵+ x⁶³=(x⁷)⁵ + (x⁷)⁹= 1⁵ + 1⁹ = 1 + 1 = 2

৪৩৬.

If x² + 2xy + y² = 25 and xy = 6, then x + y is:

ক
± 5
খ
± 10
গ
± 13
ঘ
± 4

ব্যাখ্যা

Question: If x² + 2xy + y² = 25 and xy = 6, then x + y is:

Solution:
Given that,
x²+ 2xy + y² = 25 and xy = 6
⇒ (x + y)² = 25
⇒ x + y = ± 5

৪৩৭.

What is the next number in the following sequence?
12, 41, 169, 850, ?

ক
5100
খ
5105
গ
5130
ঘ
5205

ব্যাখ্যা

Question: What is the next number in the following sequence?
12, 41, 169, 850, ?

Solution:
(12 × 3) + 5 = 41
(41 × 4) + 5 =169
(169 × 5) + 5 = 850
(850 × 6) + 5 = 5105

৪৩৮.

If n = 2.5m + 4 and 15m - 2 = 40 then n = ?

ক
22
খ
11
গ
28
ঘ
36
ঙ
None

ব্যাখ্যা

Question: If n = 2.5m + 4 and 15m - 2 = 40 then n = ?

Solution:
Given, 15m - 2 = 40
15m = 42
m = 42/15
m = 2.8

Now, n = 2.5m + 4
= (2.5 × 2.8) + 4
∴ n = 11

৪৩৯.

If the equation 4x² + x(p + 1) + 1 = 0 has exactly two equal roots, one of the value of p is-

ক
5
খ
- 3
গ
3
ঘ
0

ব্যাখ্যা

Question: If the equation 4x² + x(p + 1) + 1 = 0 has exactly two equal roots, one of the value of p is-

Solution:
Roots are equal if discriminant
b² - 4ac = 0

In given equation:
(p + 1)² - 4.4.1 = 0
⇒ (p + 1)² = 16 = (4)²
⇒ p +1 = ± 4

If take positive value
p = 3

If take negative value
p = - 5

৪৪০.

What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

ক
8.33
খ
9.33
গ
10.63
ঘ
11.63

ব্যাখ্যা

Question: What is the mean of the range, mode and median of the data given below?
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Solution:
Given data = 5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4
Arranging in ascending order = 2, 3, 3, 4, 4, 4, 5, 6, 8, 9, 9, 10, 11, 15, 19
Here,
Most frequent data is 4
So, Mode = 4

Total terms in the given data, (n) = 15 (It is odd)
∴ Median = {(n + 1)/2}^th term
= {(15 + 1)/2}^th term
= (8)^th term
= 6

Now,
Range = (Maximum value - Minimum value) + 1
= (19 - 2) + 1
= 18

∴ Mean of Range, Mode and Median = (Range + Mode + Median)/3
= (18 + 4 + 6)/3 = 28/3 = 9.33

৪৪১.

The equation of the given curve is:

ক
y = 7x
খ
x²= 9y
গ
x² + y² = 25
ঘ
f(t) = (A + Bt)e^{- Ct}

ব্যাখ্যা

The equation of the given curve is:
f(t) = (A + Bt)e^{- Ct}
y = 7x is the equation of the straight line.
x² = 9y is the equation of conic
x² + y² = 25 is the equation of circle.

৪৪২.

If ab + bc + ca = 0, then what is the value of {1/(a² - bc)} + {1/(b² - ca)} + {1/(c² - ab)}?

ক
ক) 0
খ
খ) 1
গ
গ) abc
ঘ
ঘ) 1/abc

ব্যাখ্যা

Question: If ab + bc + ca = 0, then what is the value of {1/(a² - bc)} + {1/(b² - ca)} + {1/(c² - ab)}?

Solution:
Given that
ab + bc + ca = 0

ab = - bc - ca
bc = - ab - ca
ca = - ab - bc

{1/(a² - bc)} + {1/(b² - ca)} + {1/(c² - ab)}
= {1/(a² + ab + ca)} + {1/(b² + ab + bc)} + {1/(c² + bc + ca)}
= [1/{a(a + b + c)}] + [1/{b(a + b + c)}] + [1/{c(a + b + c)}]
= (bc + ac + ab)/{abc(a + b + c)}
= 0//{abc(a + b + c)}
= 0

৪৪৩.

If x : y = 5 : 2 then the value of (xy + y²)/(x² - y²) ?

ক
2/3
খ
3/7
গ
5/9
ঘ
7/11

ব্যাখ্যা

Question: If x : y = 5 : 2 then the value of (xy + y²)/(x² - y²) ?

Solution: Let x = 5k and y = 2k

Now,
(xy + y²)/(x² - y²)
= {(5k)(2k)+(2k)²}/{(5k)² - (2k)²}
= (10k² + 4k²)/(25k² - 4k²)
= 14k²/21k²
= 14/21
= 2/3

৪৪৪.

X and Y Clubs consist of 300 and 370 members respectively. If the total member of the two clubs is 590 then how many members belong to both clubs?

ক
ক) 50
খ
খ) 60
গ
গ) 70
ঘ
ঘ) 80

ব্যাখ্যা

Question: X and Y Clubs consist of 300 and 370 members respectively. If the total member of the two clubs is 590 then how many members belong to both clubs?

Solution :
সমাধান:
ধরি
n(A) = 300 , n(B) =370 এবং n(A ∪ B) = 590

আমরা জানি
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 300 + 370 - 590
= 670 - 590
= 80

৪৪৫.

ক
12
খ
2
গ
5
ঘ
0

ব্যাখ্যা

Question:

Solution:

৪৪৬.

What is the value of 5/(x - x) = ?

ক
5
খ
0
গ
2.5
ঘ
undefined
ঙ
None of these

ব্যাখ্যা

Question: What is the value of 5/(x - x) = ?

Solution:
Given that,
5/(x - x)
= 5/0 = Undefined
Division by zero is undefined in mathematics.

৪৪৭.

If a+b = 8 and ab = 15, what is the value of a²+b²?

ক
ক) 94
খ
খ) 79
গ
গ) 34
ঘ
ঘ) 30

ব্যাখ্যা

দেওয়া আছে, a+b = 8, ab = 15
আমরা জানি, a²+b² = (a+b)²-2ab
= 8²-2×15 = 34

৪৪৮.

If Lalon loses 8 pounds, he will weigh twice as much as his sister. Together they now weigh 278 pounds. What is Lalon’s present weight, in pounds?

ক
ক) 135
খ
খ) 139
গ
গ) 147
ঘ
ঘ) 188

ব্যাখ্যা

Let, L = Lalon’s current weight, in pounds
S = Sister’s current weight, in pounds

We are told that “If Lalon loses 8 pounds, he will weigh twice as much as his sister.'' We put this into an equation:
L – 8 = 2S
∴ L = 2S + 8...... (i)

Next, we are told that “Together they now weigh 278 pounds.” We can also put this into an equation.
L + S = 278........ (ii)
To solve this equation, we can substitute 2S + 8 from Equation (i) for the variable L in Equation 2:
2S + 8 + S = 278
3S = 270
S = 90

From equation (ii) we can find,
L = 278 - 90 = 188

৪৪৯.

If p and n are integers such that p > n > 0 and p²- n²= 12, which of the following value of p - n?

ক
ক) - 1
খ
খ) 2
গ
গ) 8
ঘ
ঘ) 18

ব্যাখ্যা

p > n > 0 এবং p² - n² = 12
অপশন চেক
ধরি
p = 4 , n = 2
p² - n² = 12
4² - 2² = 12
p - n = 4 - 2 = 2

৪৫০.

The sum of first 17 terms of the series 5, 9, 13, 17, ............ is

ক
529
খ
462
গ
629
ঘ
523

ব্যাখ্যা

Question: The sum of first 17 terms of the series 5, 9, 13, 17, ............ is

Solution:
৯ - ৫ = ৪
১৩ - ৯ = ৪
∴ সাধারণ অন্তর, d = ৪
প্রথম পদ, a = ৫
পদের সংখ্যা, n = ১৭

প্রথম ১৭ পদের সমষ্টি = (n/2){2a + (n - 1)d}
= (১৭/২){২ × ৫ + (১৭ -১) × ৪}
= (১৭/২) (১০ + ৬৪)
= (১৭/২) × ৭৪
= ১৭ × ৩৭
= ৬২৯

৪৫১.

If the average of 5 consecutive number is 12, what is the sum of the least and the greatest of the integers?

ক
ক) 10
খ
খ) 24
গ
গ) 12
ঘ
ঘ) 14

ব্যাখ্যা

প্রশ্ন: If the average of 5 consecutive number is 12, what is the sum of the least and the greatest of the integers?

সমাধান:
ধরি, পাঁচটি ক্রমিক সংখ্যা x, x + 1, x + 2, x + 3, x + 4

পাঁচটি ক্রমিক সংখ্যার গড় ১২
পাঁচটি সংখ্যার সমষ্টি = ১২ × ৫
= ৬০

প্রশ্নমতে,
x + x + 1 + x + 2 + x + 3 + x + 4 = 60
⇒ 5x + 10 = 60
⇒ 5x = 50
∴ x = 10

সবচেয়ে ছোট সংখ্যাটি ১০
সবচেয়ে বড় সংখ্যাটি ১০ + ৪ = ১৪

∴সবচেয়ে বড় সংখ্যা ও সবচেয়ে ছোট সংখ্যার সমষ্টি = ১০ + ১৪
= ২৪

৪৫২.

If P^x = Q^y= R^z and Q/P = R/Q then [2z/(x + z)]³ = ?

ক
y³/x³
খ
y³/z²
গ
z/y
ঘ
x/y

ব্যাখ্যা

Question: If P^x = Q^y = R^z and Q/P = R/Q then [2z/(x + z)]³ = ?
(Officer Cash 22 এর অনুরূপ) )

Solution:
ধরি,
P^x = Q^y = R^z = k
এখন,
P^x = k
∴ P = k^(1/x)

অনুরুপভাবে,
Q^y = k
∴ Q = k^(1/y)
এবং
R^z = k
∴ R = k^(1/z)

আবার,
⇒ Q/p = R/Q
⇒ Q² = PR
⇒ {k^(1/y)}² = k^(1/x) × k^(1/z)
⇒ k^(2/y) = k^{(z + x)/xz}
⇒ 2/y = (z + x)/xz
⇒ 2xz = y(z + x)
∴ 2z/(x + z) = y/x

∴ [2z/(x + z)]³ = y³/x³

৪৫৩.

In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

ক
36
খ
45
গ
54
ঘ
65

ব্যাখ্যা

Question: In a tourist group of 100 people, 55 speak French, 40 speak Spanish, and 20 speak none of the languages. How many of them speak just one language?

Solution:

Let,
Number of people who can speak both languages = x persons
∴ Number of people who speak only French = (55 - x) persons
∴ Number of people who speak only Spanish = (40 - x) persons

Given that,
Number of people who speak none of the languages = 20 persons

According to the question,
Only French + Both + Only Spanish = Total students - Those who speak none
⇒ (55 - x) + x + (40 - x) = 100 - 20
⇒ 95 - x = 80
⇒ x = 95 - 80
∴ x = 15

∴ Only French = (55 - 15) = 40 persons
∴ Only Spanish = (40 - 15) = 25 persons

∴ Number of people who speak only one language (French or Spanish) = (40 + 25) = 65 persons

৪৫৪.

If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}, what is A ∩ B?

ক
{1, 3, 5, 7, 9}
খ
{2, 4, 6, 8}
গ
{1, 2, 3, 4, 5, 6, 7, 8, 9}
ঘ
{ }

ব্যাখ্যা

Question: If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}, what is A ∩ B?

Solution:
Given,
A = {1, 3, 5, 7, 9}
and B = {2, 4, 6, 8}

A ∩ B = {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8}
= { }

৪৫৫.

Which of the following is equivalent to (2x + 4)/(2x² + 8x + 8) for all values of x for which both expressions are defined?

ক
1/(2x² + 6)
খ
2/(x + 6)
গ
1/(x + 4)
ঘ
1/(x + 2)

ব্যাখ্যা

Question: Which of the following is equivalent to (2x + 4)/(2x² + 8x + 8) for all values of x for which both expressions are defined?

Solution:
(2x + 4)/(2x² + 8x + 8)
= {2(x + 2)}/{2(x² + 4x + 4)}
= (x + 2)/(x + 2)²
= 1/(x + 2)

৪৫৬.

If H > M = D > P and K = H > T > Z, then which of the following options is not correct?

ক
H > D
খ
Z < H
গ
P < M
ঘ
Z > K

ব্যাখ্যা

Question: If H > M = D > P and K = H > T > Z, then which of the following options is not correct?

Solution:
H > M = D > P and K = H > T > Z
By combining: Z < T< K = H > M = D > P

1) P < M True (as, M = D > P → M > P)

2) H > D True (as, H > M = D → H > D)

3) Z < H → True (as, Z < T< K = H → H > Z)

4) Z > K → False (as, Z < T< K, implies Z < K)

৪৫৭.

If y/x = 1/5 and x + 2y = 21 then x is -

ক
12
খ
15
গ
18
ঘ
24

ব্যাখ্যা

দেয়া আছে, y/x = 1/5
∴ x = 5y

এবং x + 2y = 21
⇒ 5y + 2y = 21
⇒ 7y = 21
∴ y = 3

∴ x = 5y
= 5 × 3
= 15
Answer: 15

৪৫৮.

What is the sum of the roots of the equation x² - 6x + 9 = 0.

ক
0
খ
6
গ
9
ঘ
3

ব্যাখ্যা

Question: What is the sum of the roots of the equation x² - 6x + 9 = 0.

Solution:
x² - 6x + 9 = 0
⇒ x² - 2.x.3 + 3² = 0
⇒ (x - 3)² = 0
⇒ (x - 3)(x - 3) = 0
∴ x = 3 or x = 3

∴ The sum of the roots = 3 + 3 = 6

৪৫৯.

5p - 3q = 42; 5p + 3q = 18 Given this system of equations, what is the value of |p| + |q|?

ক
ক) 2
খ
খ) 4
গ
গ) 6
ঘ
ঘ) 10

ব্যাখ্যা

5p - 3q = 42
5p + 3q = 18
____________
10p = 60
⇒ p = 6

So, Value of q = -4

∴ |p| + |q| = |6| + |-4| = 6 + 4 = 10

৪৬০.

If 9x² - qx + 16 is a square number, then q =?

ক
ক) 12
খ
খ) 18
গ
গ) 20
ঘ
ঘ) 24

ব্যাখ্যা

Question: If 9x² - qx + 16 is a square number, then q =?

Solution:
Given that,
9x² - qx + 16
= (3x)² - 2.3.4x + (4)²
= (3x)² - 24x + (4)²

∴ q = 24

৪৬১.

If x + 4/x = 4, then x/(x² + x - 1) =?

ক
1/5
খ
2/3
গ
2/7
ঘ
2/5

ব্যাখ্যা

প্রশ্ন: If x + 4/x = 4, then x/(x² + x - 1) =?

সমাধান:
দেওয়া আছে
x + 4/x = 4
⇒ (x² + 4)/x = 4
⇒ x² + 4 = 4x
⇒ x²- 4x + 4 = 0
⇒ (x - 2)² = 0
⇒ x - 2 = 0
⇒ x = 2

প্রদত্ত রাশি
x/(x² + x - 1)
= 2(2² + 2 - 1)
= 2/(4 + 2 - 1)
= 2/5

৪৬২.

If a² + b² = 45 and ab = 18, find (1/a) + (1/b)

ক
ক) 1/3
খ
খ) 1/2
গ
গ) 2/3
ঘ
ঘ) 1/4

ব্যাখ্যা

Question: If a² + b² = 45 and ab = 18, find (1/a) + (1/b)

Solution:
দেওয়া আছে,
ab = 18
a² + b² = 45
⇒ (a + b)² - 2ab = 45
⇒ (a + b)² - 36 = 45
⇒ (a + b)² = 81
∴ a + b = 9

এখন,
(1/a) + (1/b)
=(b + a)/ab
= 9/18
= 1/2

৪৬৩.

Find the value of k if (x - 1) is a factor of 4x² + 3x² − 4x + k

ক
ক) 1
খ
খ) - 3
গ
গ) 2
ঘ
ঘ) 3

ব্যাখ্যা

4x² + 3x² − 4x + k = 0
⇒ 4(1)² + 3(1)² - 4(1) + k = 0 [As, x - 1 is a factor]
⇒ 4 + 3 - 4 + k = 0
⇒ k = - 3

৪৬৪.

i^{- 49} =?

ক
0
খ
- 1
গ
- i
ঘ
i

ব্যাখ্যা

প্রশ্ন: i^-49 =?

সমাধান:
আমরা জানি, i = √-1;
i²= -1;
i³ = i²i = - i;
i⁴ = i².i² = (-1).(-1) = 1

i^{- 49}
= 1/i⁴⁹
= 1/{i⁴⁸.i}
= 1/{(i⁴)¹².i}
= 1/i
= i⁴/i
= i³
= - i

৪৬৫.

If p = a + 1/a and q = a - 1/a then p⁴ + q⁴ - 2p²q² = ?

ক
2
খ
4
গ
8
ঘ
16

ব্যাখ্যা

Question: If p = a + 1/a and q = a - 1/a then p⁴ + q⁴ - 2p²q² = ?

Solution:
p = a + 1/a
q = a - 1/a

p + q = a + 1/a + a - 1/a
= 2a

p - q =a + 1/a - (a - 1/a) = a + 1/a - a + 1/a
= 2/a

(p + q)(p - q) = (2a)(2/a) = 4
p² - q² = 4

^p4 + q⁴ - 2p²q² = (p²)² + (q²)² - 2p²q²
= (p² - q²)²
= 4²
= 16

৪৬৬.

a + b + c = 5 and ab + bc + ca = 7 find the value of a² + b² + c².

ক
22
খ
13
গ
17
ঘ
11
ঙ
None

ব্যাখ্যা

Question: a + b + c = 5 and ab + bc + ca = 7 find the value of a² + b² + c².

Solution:
Given,
a + b + c = 5
ab + bc + ca = 7

We know,
a² + b² + c² = (a + b + c)² - 2(ab + bc + ca)
= 5² - (2 × 7)
= 11

৪৬৭.

A survey in a class shows that 20 of the pupils play hockey, 18 play basketball and 8 play both hockey and basketball. How many pupils are there in the class, if everyone plays at least one of these games?

ক
18
খ
21
গ
24
ঘ
30
ঙ
42

ব্যাখ্যা

Question: A survey in a class shows that 20 of the pupils play hockey, 18 play basketball and 8 play both hockey and basketball. How many pupils are there in the class, if everyone plays at least one of these games?

Solution:
ধরি, যারা হকি খেলে তাদের সেট হলো H এবং যারা বাস্কেটবল খেলে তাদের সেট হলো B।
দেওয়া আছে,
হকি খেলে, n(H) = 20 জন
বাস্কেটবল খেলে, n(B) = 18 জন
উভয় খেলা খেলে, n(H ∩ B) = 8 জন

যেহেতু ক্লাসের সবাই অন্তত একটি খেলা খেলে, তাই ক্লাসের মোট শিক্ষার্থীর সংখ্যা হবে n(H U B)।

আমরা জানি,
n(H U B) = n(H) + n(B) - n(H ∩ B)
= 20 + 18 - 8
= 38 - 8
= 30

সুতরাং, ক্লাসে মোট শিক্ষার্থীর সংখ্যা 30 জন।

৪৬৮.

If a series looks like: 1, 1.5, 2.5, 4,........, then which of the following comes next in the sequence?

ক
ক) 5
খ
খ) 6
গ
গ) 8
ঘ
ঘ) 7

ব্যাখ্যা

ধারা : 1 1.5 2.5 4
অন্তর : .5 1 1.5
∴ পরিবর্তী সংখ্যটি (4 + 2) = 6

আবার,
1 + 1.5 = 2.5
1.5 + 2.5 = 4
∴ পরিবর্তী সংখ্যাটি = 2.5 + 4 = 6.5

প্রশ্নের অপশন কিছুটা মডিফাই করা হয়েছে।

৪৬৯.

A sum of TK. 304 was divided among 80 boys and girls in such a way that each boys gets TK. 4.5 and each girl TK. 3.5. The number of girls is-

ক
42
খ
24
গ
56
ঘ
65

ব্যাখ্যা

Question: A sum of TK. 304 was divided among 80 boys and girls in such a way that each boys gets TK. 4.5 and each girls TK. 3.5. The number of girls is-

Solution:
মনে করি,
বালিকার সংখ্যা x জন
বালকের সংখ্যা (80 - x) জন

1 জন বালিকা পায় 3.5 টাকা
x জন বালিকা পায় (3.5 × x) টাকা

1 জন বালক পায় 4.5 টাকা
(80 - x) জন বালক পায় 4.5 ×(80 - x) টাকা

প্রশ্নমতে,
(3.5 × x) + 4.5 ×(80 - x) = 304
or, 3.5x + 360 - 4.5x = 304
or, - x = 304 - 360
or, - x = - 56
or, x = 56
∴ x = 56

∴ বালিকার সংখ্যা 56 জন

৪৭০.

If y < 0 and 4x > y, which of the following could be equal to x/y?

ক
1
খ
1/2
গ
0
ঘ
1/4

ব্যাখ্যা

Question: If y < 0 and 4x > y, which of the following could be equal to x/y?

Solution:
4x > y
x > y/4
x/y < 1/4 [y < 0; y একটি ঋণাত্মক সংখ্যা]

অপশনে 1/4 চেয়ে ছোট একমাত্র সংখ্যা 0।

৪৭১.

Q. (36-55): Choose the correct answer.

In a survey, 30% of the people surveyed owned a personal computer and 75% owned a cellular telephone. If 25% owned both a cellular telephone and a personal computer, then the percentage of the people who does not have either of the instrument?

ক
ক) 40%
খ
খ) 30%
গ
গ) 25%
ঘ
ঘ) 20%

ব্যাখ্যা

Question: In a survey, 30% of the people surveyed owned a personal computer and 75% owned a cellular telephone. If 25% owned both a cellular telephone and a personal computer, then the percentage of the people who does not have either of the instrument?

Solution:
people surveyed owned a personal computer = n(P) = 30%
people surveyed owned a cellular telephone = n(C) = 75%
people surveyed owned both a cellular telephone and a personal computer = n(P ∩ C) = 25%

people surveyed owned both or anyone of them = n (P ∪ C)
= n(P) + n(C) - n(P ∩ C)
= 30% + 75% - 25%
= 80%

∴ then the percentage of the people who does not have either of the instrument = 100% - 80%
= 20%

৪৭২.

The 2^nd and 8^th term of an arithmetic progression are 17 and - 1 respectively. What is the 14^th term?

ক
- 22
খ
- 25
গ
- 19
ঘ
- 28

ব্যাখ্যা

Question: The 2^nd and 8^th term of an arithmetic progression are 17 and - 1 respectively. What is the 14^th term?

Solution:
Let the first term be a and the common difference be d.

We know,
n term of arithmetic progression = a + (n - 1)d
Then,
2^nd term, a + d = 17 ……(i)
8^th term, a + 7d = - 1 ……(ii)

Now, Subtract (i) from (ii),
(a + 7d) - (a + d) = - 1 - 17
⇒ 6d = - 18
∴ d = - 3
From (i) we get,
⇒ a + ( - 3) = 17 ; [d = - 3]
⇒ a - 3 = 17
∴ a = 20

Now, 14^th term = a + 13d
= 20 + 13( - 3)
= 20 - 39
= - 19

So the 14th term of the arithmetic progression is - 19.

৪৭৩.

If (3x + 2y) = 8 and (2x - y) = 3 find the value of 'x'.

ক
2
খ
3
গ
4
ঘ
6
ঙ
None of these

ব্যাখ্যা

Question: If (3x + 2y) = 8 and (2x - y) = 3 find the value of 'x'.

Solution:
3x + 2y = 8 ........(1)
2x - y = 3 ........(2)

From (1) + (2) × 2 we get,
3x + 2y + 4x - 2y = 8 + 6
⇒ 7x = 14
∴ x = 2

৪৭৪.

If a² + b² + c² = 16 and ab + bc + ca = 10, find the value of a + b + c.

ক
± 9
খ
± 6
গ
± 7
ঘ
± 8

ব্যাখ্যা

Question: If a² + b² + c² = 16 and ab + bc + ca = 10, find the value of a + b + c.

Solution:
a² + b² + c² = 16
ab + bc + ca = 10

We know that,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
⇒ (a + b + c)² = 16 + 2 × 10
⇒ (a + b + c)² = 36
⇒ a + b + c = √36
⇒ a + b + c = ± 6

∴ The value of (a + b + c) is ± 6.

৪৭৫.

If a² + b² = 45 and ab = 18, find (1/a) + (1/b)

ক
1/3
খ
1/2
গ
2/3
ঘ
1/4

ব্যাখ্যা

Question: If a² + b² = 45 and ab = 18, find (1/a) + (1/b)

Solution:
Given that,
ab = 18
a² + b² = 45
⇒ (a + b)² - 2ab = 45
⇒ (a + b)² - 36 = 45
⇒ (a + b)² = 81
∴ a + b = 9

Now,
(1/a) + (1/b)
=(b + a)/ab
= 9/18
= 1/2

৪৭৬.

Find the wrong term in the following series.
1200, 1188, 1164, 1116, 1020, 828, 484

ক
1200
খ
1188
গ
1020
ঘ
484
ঙ
828

ব্যাখ্যা

Question: Find the wrong term in the following series.
1200, 1188, 1164, 1116, 1020, 828, 484

Solution:
Given series:
1200, 1188, 1164, 1116, 1020, 828, 484
The series decreases with multiples of 12, doubling each time.
1^st term = 1200
2^nd term = 1200 - 12 = 1188
3^rd term = 1188 - 24 = 1164
4^th term = 1164 - 48 = 1116
5^th term = 1116 - 96 = 1020
6^th term = 1020 - 192 = 828
7^th term = 828 - 384 = 444

The series has 484 as the last term, but it should be 444.

Hence the wrong term in the series is 484.

৪৭৭.

If √x = √7 - √5 then the value of x² - 24x + 8 = ?

ক
5
খ
4
গ
2
ঘ
0

ব্যাখ্যা

Question: If √x = √7 - √5 then the value of x² - 24x + 8 = ?

Solution:
Given,
√x = √7 - √5
⇒ x = 7 + 5 - 2.√7.√5 (Squaring both sides)
⇒ x = 12 - 2√35
⇒ x - 12 = - 2√35
⇒ x² + 144 - 24x = 140 (Squaring both sides)
⇒ x² + 4 - 24x = 0
⇒ x² + 8 - 24x = 4
∴ x² - 24x + 8 = 4

৪৭৮.

If 6x - y = 1 and 3x + 2y = 13, then value of (x, y) =?

ক
(1, 5)
খ
(1, 3)
গ
(4, 5)
ঘ
(2, 1)

ব্যাখ্যা

Question: If 6x - y = 1 and 3x + 2y = 13, then value of (x, y) =?

Solution:
দেওয়া আছে
6x - y = 1 ..................... (1)
3x + 2y = 13 ............... (2)

(1) × 2 + (2) ⇒
12x - 2y + 3x + 2y = 2 + 13
⇒ 15x = 15
∴ x = 1

(1) ⇒
6x - y = 1
⇒ 6 × 1 - y = 1
⇒ 6 - y = 1
⇒ - y = 1 - 6
⇒ - y = - 5
∴ y = 5

৪৭৯.

এর মান কত?

ক
(x + 7)/x
খ
(x + 4)/x
গ
(x + 5)/x
ঘ
(x - 5)/x
ঙ
কোনটি নয়

ব্যাখ্যা

প্রশ্ন:
এর মান কত?

সমাধান:

৪৮০.

In a certain population group, 57% of the people can play Cricket and 63% can play Football. If every people in the group can play at least one of the two sports, then what percent of the people can play both Cricket and Football?

ক
20%
খ
18%
গ
12%
ঘ
6%

ব্যাখ্যা

Question: In a certain population group, 57% of the people can play Cricket and 63% can play Football. If every people in the group can play at least one of the two sports, then what percent of the people can play both Cricket and Football?

Solution:
n(C) = 57%
n(F)= 63%
n(C ∪ F) = 100%

We know that,
n(C ∪ F) = n(C) + n(F) - n(C ∩ F)
⇒ n(C ∩ F) = n(C) + n(F) - n(C ∪ F)
= 57% + 63% - 100%
= 120% - 100%
= 20%

৪৮১.

If 3√5 + √125 = 17.88, then what will be the value of √80 + 6√5 = ?

ক
12.34
খ
21.33
গ
24.87
ঘ
22.35

ব্যাখ্যা

3√5 + √125
= 3√5 + √(5 × 25)
= 3√5 + 5√5
= 8√5

∴ 8√5 = 17.88
⇒ √5 = 17.88/8 = 2.235

Now, √80 + 6√5
= √(5 × 16) + 6√5
= 4√5 + 6√5
= 10√5
= 10 × 2.235
=22.35
-----------------------------------------
Alternative way:
√80 + 6√5
= √(5 × 16) + 6√5
= 4√5 + 6√5
= 10√5
= 10 × 2.236
= 22.36 ≈ 22.35 which is in option d)

৪৮২.

If a = 5 + √2, then find the value of a²?

ক
ক) 5 + 10√2
খ
খ) 20 + 5√2
গ
গ) 27
ঘ
ঘ) 27 + 10√2

ব্যাখ্যা

Question: If a = 5 + √2, then find the value of a²?

Solution:
a = 5 + √2
a² = (5 + √2)²
= 5² + 2 × 5 × √2 + (√2)²
= 25 + 10√2 + 2
= 27 + 10√2

৪৮৩.

2x² - xy - 6y² এর একটি উৎপাদক -

ক
3x - 2y
খ
3x + 2y
গ
2x - 3y
ঘ
2x + 3y

ব্যাখ্যা

2x² - xy - 6y²
= 2x²- 4xy + 3xy -6y²
= 2x(x - 2y) + 3y(x - 2y)
= (x - 2y)(2x + 3y)

৪৮৪.

If a² - 2a = 1, then =?

ক
2
খ
12
গ
14
ঘ
16

ব্যাখ্যা

Question: If a² - 2a = 1, then =?

Solution:
Given that,
a² - 2a = 1
⇒ a² - 1 = 2a
∴ a - 1/a = 2

Now,
a³ - 1/a³
= (a - 1/a)³+ 3.a.(1/a)(a - 1/a)
= (2)³ + 3 × 2
= 8 + 6
= 14

৪৮৫.

The next term of the series: 36, 81, 144, 225, ____ is

ক
300
খ
324
গ
354
ঘ
388

ব্যাখ্যা

Question: The next term of the series: 36, 81, 144, 225, ____ is

Solution:
Given, 36, 81, 144, 225,
The series is = 6² , 9², 12², 15², 18²
So, next term is 18² = 324

৪৮৬.

Nafees completes 1/4 of his thesis in 3 days. How many more days will it take to finish his thesis?

ক
ক) 12 days
খ
খ) 10 days
গ
গ) 9 days
ঘ
ঘ) 7 days

ব্যাখ্যা

Question: Nafees completes 1/4 of his thesis in 3 days. How many more days will it take to finish his thesis?

Solution:
Nafees completes 1/4 of his thesis in 3 days
Nafees completes his thesis in 3 × 4 days
= 12 days

he will need 12 - 3 = 9 days to finish his task.

৪৮৭.

If m = 7 - 4√3, then √m + 1/√m =?

ক
ক) 3
খ
খ) 4
গ
গ) 6
ঘ
ঘ) 8

ব্যাখ্যা

Question: If m = 7 - 4√3, then √m + 1/√m =?

Solution:
Given that,
m = 7 - 4√3
⇒ m = 4 - 4√3 + 3
⇒ m = (2)² - 2 . 2 . √3 + (√3)²
⇒ m = (2 - √3)²
∴ √m = 2 - √3

∴ 1/√m = 1/(2 - √3)
= {1(2 + √3)}/{(2 - √3)(2 + √3)}
= (2 + √3)/{2² - (√3)²}
= (2 + √3)/(4 - 3)
= 2 + √3

∴ √m + 1/√m = 2 - √3 + 2 + √3
= 4

৪৮৮.

Find the solution of the inequality |2x - 3| ≤ 1.

ক
[0, 1]
খ
[1, 2]
গ
[2, 3)
ঘ
(0, 2]

ব্যাখ্যা

Question: Find the solution of the inequality |2x - 3| ≤ 1.

Solution:
Given that,
|2x - 3| ≤ 1
⇒ - 1 ≤ 2x - 3 ≤ 1
⇒ - 1 + 3 ≤ 2x ≤ 1 + 3 ; [adding 3 to all parts]
⇒ 2 ≤ 2x ≤ 4
⇒ 1 ≤ x ≤ 2 ; [dividing all parts by 2]

Therefore, the solution is 1 ≤ x ≤ 2 or in interval notation [1, 2]

৪৮৯.

If p = 3 + √2 then, Find the value of p².

ক
7 + 3√2
খ
5 + 7√2
গ
11 + 6√2
ঘ
7 + 9√2

ব্যাখ্যা

Question: If p = 3 + √2 then, Find the value of p².

Solution:
Given,
p = 3 + √2
⇒ p²= (3 + √2)²
= 3² + 2 · 3 · √2 + (√2)²
= 9 + 6√2 + 2
= 11 + 6√2

৪৯০.

If 2x + y = 7 and x - 2y = 1 then (x, y) = ?

ক
(2, 3)
খ
(3, 1)
গ
(4, 2)
ঘ
(5, 2)

ব্যাখ্যা

Question: If 2x + y = 7 and x - 2y = 1 then (x, y) = ?

Solution:
2x + y = 7 .............(1)
x - 2y = 1 .............(2)
(1) × 1 + (2) × 2 ⇒
2x + y + 2x - 4y = 7 + 2
⇒ 4x - 3y = 9
⇒ 4x = 9 + 3y ...........(3)
(3) এর মান (1) নং সমীকরণে বসিয়ে পাই,
2x + y = 7
⇒ 2(9 + 3y)/4 + y = 7
⇒ (18 + 6y)/4 + y = 7
⇒ 18 + 6y + 4y = 28
⇒ 10y = 10
⇒ y = 1
y এর মান (3) নং সমীকরণে বসিয়ে পাই,
4x = 9 + 3(1)
⇒ 4x = 12
⇒ x = 3
নির্ণেয় সমাধান (x, y) = (3, 1)

৪৯১.

If x > 2 and x < 3, then which of the following is positive?
(I) (x - 2)(x - 3)
(II) (2 - x)(x - 3)
(III) (2 - x)(3 - x)

ক
(I) only
খ
(III) only
গ
(I) and (II) only
ঘ
(II) only

ব্যাখ্যা

Question: If x > 2 and x < 3, then which of the following is positive?
(I) (x - 2)(x - 3)
(II) (2 - x)(x - 3)
(III) (2 - x)(3 - x)

Solution:
So x is between 2 and 3.

X is more than 2, so x - 2 will be positive.
X is less than 3, so x - 3 will be negative.
∴ (x - 2)(x - 3), then, is a positive times a negative, and is thus negative.

X is more than 2, so 2 - x will be negative.
We already know that x - 3 is negative,
So (2 - x)(x - 3) is a negative times a negative, and is thus definitely positive.

X is less than 3, so 3 - x will be positive.
and 2 - x is negative.
So (2 - x)(3 - x) is a negative times a positive, which is negative.

We can see then, that only (2 - x)(x - 3) is positive.
The answer is II only.

৪৯২.

If (3 + √3)z + 2 = 5 + 3√3, then the value of z is-

ক
ক) √3
খ
খ) √3 + 3
গ
গ) 1/√3
ঘ
ঘ) 2√3 + 3

ব্যাখ্যা

Question: If (3 + √3)z + 2 = 5 + 3√3, then the value of z is-

Solution:
(3 + √3)z + 2 = 5 + 3√3
⇒ (3 + √3)z + 2 = 5 + 3√3
⇒ (3 + √3)z = 5 + 3√3 - 2
⇒ (3 + √3)z = 3 + 3√3
⇒ z = (3 + 3√3)/(3 + √3)
⇒ z =√3(3 + √3)/(3 + √3)
z = √3

৪৯৩.

If a³ = 117 + b³ and a = 3 + b, then the value of a + b is?

ক
ক) 7
খ
খ) 13
গ
গ) 49
ঘ
ঘ) 0

৪৯৪.

The factors of 4x⁴ + 1 is -

ক
ক) (2x² + 2x - 1) (2x² - 2x + 1)
খ
খ) (2x² + 2x + 1) (2x² - 2x + 1)
গ
গ) (2x² + 2x - 1) (2x² - 2x - 1)
ঘ
ঘ) (2x² + 2x + 1) (2x² - 2x - 1)

ব্যাখ্যা

4x⁴ + 1
= (2x²)² + 1
= (2x²)² + 2.2x².1 + 1² - 4x²
= (2x² + 1)² - (2x)²
= (2x² + 2x + 1) (2x² - 2x + 1)

৪৯৫.

ক
4/3
খ
5/2
গ
6/5
ঘ
9/4
ঙ
2/3

৪৯৬.

Find the domain of f(x) = 1/(x + 3)

ক
- 3
খ
3
গ
x ≠ - 3
ঘ
R - {- 3}

ব্যাখ্যা

Question: Find the domain of f(x) = 1/x + 3

Solution:
দেওয়া আছে,
f(x) = 1/x + 3

আমরা জানি,
একটি ভগ্নাংশের হর (denominator) শূন্য হতে পারবে না।
অর্থাৎ,
⇒ x + 3 ≠ 0
∴ x ≠ - 3

সুতরাং, f(x) এর ডোমেইন = R - {- 3}

৪৯৭.

ক
ক) 2
খ
খ) 3
গ
গ) 5
ঘ
ঘ) 7

ব্যাখ্যা

Question:

Solution:
{³√(2⁴)⁶} - 1
= (2⁴)^6/3 - 1
= (2⁴)² - 1
= (2⁴ + 1) (2⁴ - 1)
= 17 × 15
= 17 × 3 × 5

So, the smallest prime factor is 3

৪৯৮.

If x - 1/x = √5, then x + 1/x =?

ক
√3
খ
3√3
গ
1/3
ঘ
3

ব্যাখ্যা

Question: If x - 1/x = √5, then x + 1/x =?

Solution:
Given,
x - 1/x = √5

We know,
(x + 1/x)² = (x - 1/x)² + 4x.1/x
⇒ (x + 1/x)² = ( √5)² + 4
⇒ (x + 1/x)² = 5 + 4
⇒ (x + 1/x)² = 9
⇒ (x + 1/x) = √9
∴ (x + 1/x) = 3

৪৯৯.

If x > 7 and y > - 3 then which of the following is true?

ক
ক) - x > 2y
খ
খ) xy < - 21
গ
গ) xy > - 21
ঘ
ঘ) x > - 2y

ব্যাখ্যা

Question: If x > 7 and y > - 3 then which of the following is true?

Solution:
Given,
x > 7 and y > - 3
Or, xy > 7 × (- 3)
Or, xy > - 21

৫০০.

If a² + b² = 45 and ab = 18, find (1/a) + (1/b).

ক
1/3
খ
1/2
গ
2/3
ঘ
1/4
ঙ
None of these

ব্যাখ্যা

Question: If a² + b² = 45 and ab = 18, find (1/a) + (1/b).

Solution:
দেওয়া আছে,
ab = 18
a² + b² = 45
⇒ (a + b)² - 2ab = 45
⇒ (a + b)² - 36 = 45
⇒ (a + b)² = 81
∴ a + b = 9

এখন,
(1/a) + (1/b)
= (b + a)/ab
= 9/18
= 1/2